A Class of Polynomial Modules over Map Lie Algebras

For any finitely generated unital commutative associative algebra \(\mathcal {R}\) over \(\mathbb {C}\) and any complex finite-dimensional simple Lie algebra \(\mathfrak {g}\) with a fixed Cartan subalgebra \(\mathfrak {h}\), we classify all \(\mathfrak {g}\otimes \mathcal {R}\)-modules on \(U(\mathfrak {h})\) such that \(\mathfrak {h}\) as a subalgebra of \(\mathfrak {g}\otimes \mathcal {R}\), acts on \(U(\mathfrak {h})\) by the multiplication. We construct these modules explicitly and study their module structures.